Final answer:
The linear approximation of the function f(x, y) = y + x at the point (1, 3) is found using the tangent plane approximation and simplifies to f(x, y) ≈ x + y, which is identical to the original function, confirming that the function is linear.
Step-by-step explanation:
The question asks to find the linear approximation of the function f(x, y) = y + x at the point (1, 3). To find the linear approximation, also known as the tangent plane approximation in multivariable calculus, we look for an equation in the form f(x, y) ≈ f(a, b) + fx(a, b)(x - a) + fy(a, b)(y - b), where fx and fy are the partial derivatives of f with respect to x and y, respectively, and the point (a, b) is where the approximation is centered—in this case, (1, 3).
The function simplifies to f(x, y) = x + y. The partial derivatives are fx = 1 and fy = 1. Plugging the coordinates of the point (1, 3) into the approximation formula gives us:
f(x, y) ≈ f(1, 3) + fx(1, 3)(x - 1) + fy(1, 3)(y - 3)
f(x, y) ≈ 4 + (1)(x - 1) + (1)(y - 3)
f(x, y) ≈ 4 + x - 1 + y - 3
f(x, y) ≈ x + y.
This simplification shows that the linear approximation of the function f(x, y) at the point (1, 3) is the same as the original function. This is because the function is already linear in its variables x and y.